Properties

Label 1.6.K.24.12b
  
Name \(\mathrm{SU}(2)\times O\)
Weight $1$
Degree $6$
Real dimension $4$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)\times\mathrm{U}(1)_2\)
Component group \(S_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$24$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{SU}(2)\times\mathrm{U}(1)_2$
$\mathbb{R}$-dimension:$4$
Description:$\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$

Component group

Name:$S_4$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{SU}(2)\times D_4$, $\mathrm{SU}(2)\times T$, $\mathrm{SU}(2)\times D_3$
Minimal supergroups:$\mathrm{SU}(2)\times J(O)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $26$ $0$ $345$ $0$ $5824$ $0$ $116004$ $0$ $2611356$
$a_2$ $1$ $2$ $9$ $51$ $358$ $2902$ $26245$ $258575$ $2727940$ $30396804$ $353744551$ $4260593031$ $52727046564$
$a_3$ $1$ $0$ $12$ $0$ $810$ $0$ $106345$ $0$ $20169478$ $0$ $4778002866$ $0$ $1291783829820$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $5$ $14$ $26$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $51$ $31$ $91$ $56$ $174$ $345$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $71$ $358$ $216$ $134$ $689$ $420$ $1378$ $835$ $2810$ $5824$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $531$ $2902$ $322$ $1735$ $1046$ $5878$ $3514$ $2114$ $12164$ $7245$ $25523$ $15134$ $54152$ $116004$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $810$ $4473$ $26245$ $2676$ $15482$ $9180$ $55078$ $5463$ $32402$ $19133$ $117150$ $68654$ $40386$ $251669$
$$ $146944$ $545174$ $317184$ $1189440$ $2611356$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&3&0&5&9&0\\1&0&6&0&4&0&9&9&0&4&0&12&0&0&20\\0&2&0&5&0&8&0&0&8&0&3&0&12&14&0\\1&0&4&0&7&0&9&10&0&7&0&16&0&0&24\\0&6&0&8&0&26&0&0&28&0&14&0&36&52&0\\1&0&9&0&9&0&24&21&0&13&0&35&0&0&64\\2&0&9&0&10&0&21&29&0&16&0&37&0&0&72\\0&6&0&8&0&28&0&0&40&0&16&0&46&70&0\\1&0&4&0&7&0&13&16&0&17&0&24&0&0&52\\0&3&0&3&0&14&0&0&16&0&13&0&23&34&0\\1&0&12&0&16&0&35&37&0&24&0&72&0&0&124\\0&5&0&12&0&36&0&0&46&0&23&0&76&98&0\\0&9&0&14&0&52&0&0&70&0&34&0&98&151&0\\2&0&20&0&24&0&64&72&0&52&0&124&0&0&260\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&7&26&24&29&40&17&13&72&76&151&260&116&123&254&191&56\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.